Sunday, January 01, 2012
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The most urgent challenge facing mankind is that every 3,200 years, we move Spring by one day :-)

Happy New Year! The United Nations declared 2012 to be the International Year of Cooperatives (a communist-style collective ownership format) and the International Year of Sustainable Energy For All (a contradiction with the energy conservation law) which are so stupid labels that I won't bother to discuss them.

After four years, we are entering a leap year. Just to be sure, why do we have leap years?

The most regular unit of time that people have used was a solar day – roughly speaking, it should be the period of time between two noons. The idea is that when your clocks add 1 solar day, you return to the same phase of the day. The solar day was divided to \(24\times 3,600=86,400\) seconds.

Today, we define one second much more accurately, using atomic periods used in atomic clocks. That's why we are able to detect irregularities in the motion of the Earth and the celestial bodies. But you shouldn't forget that the new definition of one second was chosen to agree with \(1/86,400\) of a solar day.

Solar day vs sidereal day

The solar day is mostly determined by the rotation of the Earth around its axis. However, this rotation has to be measured relatively to the Sun, i.e. in a rotating reference frame where the direction Earth-Sun is kept approximately constant. The period after which the line Prague-Boston points to the same star isn't 24 hours. It's shorter. How much shorter?

Well, the frequency of the Earth's orbital motion around the Sun is approximately
\[ f_S= 1/365.25\,\,{\rm revolutions\,\,per\,\,solar\,\,day}, \]a number we will clarify later. This low (slow) frequency must be added to the frequency of solar days
\[ f_{D} = 1\,\,{\rm rotation\,\, per\,\, solar\,\, day} \] and you get approximately
\[ f_A = f_D + f_S = 1.0027 \,\,{\rm rotations\,\, per\,\, solar\,\, day} \] The relative sign was "plus" because almost everything in the Solar System is rotating in the same direction and we are literally adding the Earth's spinning and the Sun's orbiting around the Earth, if you allow me to use the geocentric frame for a while. ;-)

The inverse of this constant is \(0.9973\) solar days per astronomical day and tells you that the astronomical day is \(0.9973\) solar days which – verify it – is 23 hours, 56 minutes, 4 seconds. So this complicated less-than-a-day, known as a sidereal day, is the periodicity after which the direction of the Earth returns to the same position relatively to the stars.

Tropical, sidereal, and anomalistic years

At the beginning of the text, you should have understood that we have a very accurate definition of one second that is meant to be very close to \(1/86,400\) of a solar day. What is the year and how long is it?

It depends what year you are talking about.

We usually mean the most practically important periodicity, namely the periodicity between two springs, or two solstices etc. Such a year is given by the periodicity (between two solstices) after which the Earth's spin axis returns to the maximum deviation away from the Sun on the Northern Hemisphere – which we want to call "December 21st", plus minus one day, forever, so that December remains colder than July on the Northern Hemisphere forever.

This period between two solstices is called a tropical year and it equals \(365.24219\) solar days which is, if you need to know, 365 days, 5 hours, 48 minutes, 45 seconds. The tropical year will be important for our discussion of the leap years below.

However, before I discuss the leap years, I must mention that there are two other important "mutations of a year" related to astronomy which don't involve the Moon. (There are many more "types of a year" if you allow them to depend on the position of the Moon.) These two extra years are the sidereal year and the tropical year.

The sidereal year is the period after which the Sun-Earth direction points to the same direction relatively to the stars. It is equal to \(365.256363004\) solar days. Note that it is longer than the tropical (solstice-related) year. The difference boils down mostly to the precession of the Earth's axis. The axis of Earth's spin isn't quite pointing to the same star all the time. Every 26,000 years or so, the axis makes a full round trip, like a spinning top. So the sidereal year is longer than the tropical year, by \(365.25/26,000=0.014\) solar days and indeed, \(365.242+0.014=365.256\) solar days.

The anomalistic year is the year between two perihelia. The Earth's orbit is elliptical but because of various corrections, the position on the closest approach to the Sun – the perihelium – isn't quite constant, either. Jupiter and other bodies are perturbing the shape of the ellipse, much like corrections from general relativity (which are much smaller for the Earth, but they are important enough for Mercury, if you care, which is why the precession of Mercury's perihelion became an early confirmation of general relativity). As a result, the anomalistic year is close to the sidereal year but it is a slightly longer again, \(365.259636\) solar days or so.

Note that each of these numbers is known rather accurately but it doesn't make much sense to try to be much more accurate than that because if you study the motion of all the celestial bodies with an even better accuracy, you find out that the motion isn't really periodic, the axis is changing in other ways than just precession, and so on.

If you missed the 2011-2012 fireworks, this fresh 15-minute video from Prague may fill your cultural gap. ;-)

Tropical year vs leap years

Let me return to the time between two solstices of the same type, the tropical year equal to \(365.24219\) solar days. You see that the number is close to \(365\) and one quarter. How do we (or how did we) design our calendar so that the winter solstice remains near December 21st?

Clearly, if we approximate the number by \(365.25\), there is an easy solution: most years should have 365 days but each fourth year should become a leap year with 366 days (February 29th is where we are adding it). In this convention, the average year has
\[ \frac 34 \times 365 + \frac 14 \times 366 = 365.25 \] solar days. That's not accurate enough: we want \(365.24219\) days. To replace \(0.25\) by \(0.24\), we must erase \(1/25\) of the leap years. It means that every year that is divisible by 100 will not be a leap year even though it is a multiple of 4. In this choice, the average year is
\[ \frac{76}{100}\times 365 + \frac{24}{100} \times 366 = 365.24 \] because every 100 years, 76 years are normal and 24 years are leap years. We erased one leap year each century: the years 1700, 1800, 1900 were not leap years even though they were multiples of 4. However, \(365.24\) is too short. So we must add some leap years again. It's good to neutralize the rule for some of the multiples. Note that our last correction was about \(4/3\) times larger than the best one. So it's good to return \(1/4\) of the freshly removed leap years.

The choice is that the years that are multiples of 400, like 1600 or 2000 (a leap year we remember well), are leap years even though they're multiples of 100 most of which were removed from the list of leap years. With this 4-100-400 convention, we get the average year's length equal to
\[ \frac{300+4-1}{400} \times 365 + \frac{100-4+1}{400}\times 366 = 365.2425 \] which is pretty close to \(365.24219\) solar days.

This brings me to the punch line. You see the catastrophe. In 3,200 years, our calendar inserts one leap year too many. So the solstice will be December 20th after 3,200 years, December 19th after 6,400 years, and so on. Around the year 600,000, the winter solstice would occur in June instead of December. This is clearly unacceptable! :-)

Note that the time after which the discrepancy grows by one day is almost exactly 3200 years. There is an easy fix. Eliminate the leap year status of some leap years, namely the years that are multiples of 3200!

So the years 3200 AD, 6400 AD, 9600 AD etc., despite their being a multiple of 400 which qualifies them to be leap years according to the currently valid rules, will not be leap years according to the proposed rules. Here is the petition:

Hi Ban Ki-moon,

on behalf of the living and future generations, we demand the years divisible by 3200, namely 3200 AD, 6400 AD, 9600 AD, and so on, will not be leap years. That will bring the average year, currently at 365.2425 solar days, to a new value of
\[ \begin{align} \dots &= \frac{2400+32-8+1}{3200}\times 365 +\\
&+\frac{800-32+8-1}{3200}\times 366\\ &= 365.2421875 \end{align} \]closer to the tropical year which is 365.24219 days.

You've been solving similar "problems" throughout your life so you seem to be the right recipient of this petition. Without the new Motlo-Gregorian calendar, the Northern Hemisphere will experience a horrible climate change: in half a million years, December will look like a June looks today, assuming that we're not in an ice age at that moment! ;-)

We also insist that the United Nations immediately stop its support for terrorists, global warming alarmists, and similar groups with goals that are detrimental for the civilization.

Yours
Population of Earth, signed by its own hand

You may sign it. ;-) If the petition succeeds, the timescale after which the discrepancy grows by one day will jump from 3200 years to hundreds of thousands of years. Also, it will finally become possible to treat the IPCC members as terrorists. ;-)

More seriously, the planning of the leap years for hundreds of thousands of years – and maybe even for 10,000 years – cannot quite be done because it's plausible that even the length of the tropical year will differ from the present one by an important enough amount that probably can't be quite predicted today (even though I am a bit puzzled about this skeptical statement).